For once, the order of the achievements you get actually kind of matters. Every achievement you get gives you a skill you can use for your runs, and some skills really help you get other achievements. That being said, here's what you should do.

Get it first, because it's super easy, and helpful.
Literally just die ten times. You can just walk into enemies, and die. Easiest for sure.

Get this next, to help with number 3.
Exactly what it sounds like. Before going into a boss battle, make sure you only have one heart, then beat the boss. The first boss is really easy; I actually got this achievement on my very first boss battle.

Most useful skill in the game. Lets you start with random weapons, making everything else much easier.
This is ONLY for a level, NOT a whole campaign, so it's not nearly as bad as it sounds. Furthermore, if you use the skill "Big Game Hunter", it'll make the boss battle 1/4 shorter, which is helpful.

This one is a bit trickier, but the skill you get is absolutely amazing (all melee weapons become ranged), especially when combined with "Born to Kale", and even more amazing when combined with "360 Blowscope". You'll definitely want this early, since it helps so much.
Some enemies will shoot projectiles at you. If you use your melee weapon at the right time, you can hit the projectiles away from you. Not only is this helpful in defense (just constantly spam melee, and you rarely get hit by bullets), but it also acts offensively, because now the projectile can hit enemies. The trick for this achievement is that after you hit the projectile with your melee weapon, the bullet HAS to hit the one that shot it, or else it doesn't count towards the achievement. For example, if you find two blueberries (little guys that shoot the single alien bullet), and one shoots at you, and you hit the bullet into the OTHER blueberry, killing him, your counter won't go up. The one that shot the bullet is the one that has to die by THAT bullet.
This achievement is easiest if you kill all the veggies other than the ranged ones, and then hit back the projectiles with a melee weapon that is NOT ranged (because you don't want to accidently kill him with your ninja star or something). The precision can be a bit hard for making it hit them. I recommend moving up or down once they shoot, so you're more in line with them.

This is another awesome skill (when you hit three enemies at once, it explodes). You'll want this early, again, because it helps make the game a lot easier. When combining this with "Born to Kale" and "Boomerang", you get a ridiculous combo where you can throw a massive sword a very far distance, instantly killing most enemies, and then blowing up, killing even more enemies, with just one tap of a button. Super OP in all honesty.
This one is another tricky one, but it's REALLY helpful if you have "Born to Kale" already. Basically, start the game, and if you don't have a grendate in your item slot, give up and start again, until you do have a grenade. With grenade in hand, run around until you get a nice little collection of enemies following you, then, quickly turn around, throw the grenage and IMMEDIATELY hit it with your melee weapon (don't even wait for it to land, throw the grenade then right away hit it). The grenade will then blow up, and should kill many of the veggies. Just do this a few times, and you've got it. Once you get the idea (and the "Born to Kale" skill), this is actually really easy to get.

The easy ones. You can probably just skip this section.

Two ways to collect RadRocks. Beat bosses, or survive tons of waves in the Synthwave. Bosses are definitely the easiest way to farm RadRocks.
If you don't know how to get to bosses, just keep following the pink arrows ON THE EDGE of the screen. Houses don't matter at all (just for looting). Find the edge of the screen with pink arrows, and beat all the baddies in there. Once you do that a few times, a guy in a little book type ship thing shows up. Beat him, and you get the boss key for that area.

Same as the Radstone Gander. Just collect tons of RadRocks, then go to the shop and buy the thing in the middle of the room that costs Radrocks. Once you get to 100 RadRocks earned, you should have more than enough to buy 5 weapons.

Misleading. This is breaking objects in game, not enemies (in-game description makes it sound like enemies, it's not). Just everywhere you go, smash everything. Helpful actually, because that's how you find ammo and keys as well.

Again, just what it sounds like. Just beat 10 bosses. Doesn't have to be different bosses (which is good, because right now there aren't 10 bosses). Like the Radstone Gander achievement, following the pink arrows on the edge of the screen, and you'll eventually get to the boss.

These achievements require beating the whole campaign in order to get them. Helpfully, all three of them can be gotten in one run! (Just in case you're skeptical, I got all three in one run, I believe on my second or third try, I just had a bit of trouble with the last boss).
Here's the rules to follow to make this possible.
Equip "Born to Kale", "360 Blowscope", and "Boomerang", so that your melee attack is absolutely devastating, slaughtering everything on the screen with one tap of a button.
Start your campaign, and if you don't have a good melee weapon, give up, and start over. I like the massive sword personally, but the pickaxe is good as well. The sledgehammer, while powerful, is a bit slow for my liking. Really, you don't even need to worry about items (I did my run without using items, just in case), but if you want, you can try to get a good melee and item for your campaign.
Only ever use your melee weapon. But, you know, it's so OP now with the skills, you only really need your melee weapon.
Ignore all houses. Never go in a house. Houses only give loot, and you've already got your good gear, so they're useless, and waste time. Since you don't go in any houses, you won't even have the option of the WeaponWheel, making it easy to resist. No houses. Seriously.
The moment you get out of your car, just run to the pink arrows, and kill all baddies in the areas off the main map. You can kill stuff as you go through the main map, but just the stuff that's in your way. Thankfully, veggies despawn when you zone, so you don't really need to kill anything except what's in your way.
After you follow all the pink arrows, and kill all the baddies in the side areas, run right to the boss, and kill it quickly. Try hitting multiple enemies at once, as often as possible, to get as many explosions as possible, so the boss will die faster.
Eat a cookie, you've got it. My run was just over 14 min, and I think it was my second try, so this achievement is very doable.

Note: If you're having trouble with the last boss, and he's ruining your runs, "Tongue of Steal" can help with the him. One of the worst parts is all the acid on the boss map, so with this skill you don't have to worry about that.

All the little random other achievements.

With "Born to Kale" equipped, just keep starting over until you get the Molotov Cocktail. Then run around until you get tons of veggies chasing you, and throw a molotov in the direction you're running, and just run past it. You can just stand there, with a pool of fire in-between you and the veggies, and just watch them run through it, and die. For the larger veggies, just shoot them to save time. It's the little guys that fire kills quickly.

This one is easier than it seems. With full hearts, open a medkit healthbox thing, so that it turns into a ton of coins. Do this 20 times.
There ARE medkit healthbox things in the world you get from opening chests, HOWEVER you don't have to rely on those! The easiest way to get this, is to make tons of money (possibly through a SynthWave Dimension), then go to the shop, and just keep buying the medkit over and over. Once they get really expensive, just start a new game, and do the process over again.
This DOES count towards the achievement, however, if you look at your computer in the shop, it won't show the progress. You need to zone, then go back in the shop or main menu, and then check your progress, and it should be updated. You can do multiple medkits at one time, and all will count.

Exactly what it sounds like. Every door on the map, for each of the three maps, needs to be unlocked. The hard part of this is getting all the keys. On each of the three maps, remember to buy the key in the shop for that map. Houses usually contain keys (sometimes even three keys!), so break all the objects in all the rooms, and all the objects on the map as well, in order to find tons of keys. If you do this, you shouldn't have a problem.

This is a random event for the first boss, where they replace the Broccoliath, same as the Shroominator is for the second boss encounter (which doesn't have an achievement). As you play the game, you'll likely find this random boss in the first level's boss area. Just kill it. The boss itself is quite easy. The coconut just lays on the ground doing nothing, so just unload on it. Then it vomits, so avoid the vomit. Kill the three dudes that show up, and unload on the coconut again. Quite simple.

When you spawn in the world, go down to the bottom left of the screen where the big hole is. Get a bunch of veggies to chase you around the hole, and then use your melee weapon to shove them in the hole. Some melee weapons have a lot more knockback, while some have none. Even just using your police stick (default melee weapon) you should be able to do this easy enough. The apples are super easy to knock in, so you can just get a run where apples spawn, and keep tossing them in there.

As your car appears on the screen, hold one of the directional buttons before you even get out, and make sure that the ENTIRE LEVEL you are ALWAYS holding a directional button, and moving. Never stop running. Make sure you don't bump into something, where it'll make you stop moving. I got this on my first try, so I don't think it's too difficult to get if you know how.

This one is annoying... One thing to note, if you break things in-between the houses, like in the main map, that DOES ruin your streak (I know from experience...). It's a very delicate procedure to do this. I used only my pistol, and I would lure enemies to areas where I could shoot without endangering objects. This will likely take a good few tries, because you'll make mistakes, and likely just have bad luck. Keep trying until you get good luck with easier buildings.

So guns do not reload in this game, so what this achievement means is that, if you have 100 ammo for your shotgun, you need to waste 99 ammo, and then kill things when your ammo is at zero.
Note = Explosives do NOT count. Your rocket launcher will NOT count towards this achievement.
Personally I found the sniper rifle to be the best, since it has huge piercing, and you can easily kill five veggies in one shot.

Keep starting new games with "Born to Kale", until you get a weapon like the grenade launcher, or grenades, or something else explosive. Then get down to 1 health, and as the veggies come at you, throw your grenade and immediately die. You should die, then it'll blow up, and you get the achievement.

I left the hardest for last. I tried to get this for probably three hours or more, and I believe I've learned some good stuff to share.
Note = My research is entirely anecdotal, meaning, this is my experience, so my conclusions "may" very well be false. That said, this worked real good for me.
There are two problems with this achievement. Firstly, how do you get to the dimension? And secondly, how do you last 20 waves?

How do I get to the SynthWave World?
Here's my experience. I start playing the game, and I'm getting the dimension quite often. As I played the game, I got it less and less. Finally, when I've done everything else, I literally sat there, grinding veggies, for AN HOUR AND A HALF, and didn't get even ONE dimension run... I think I might have figured out the problem.
When you start the game, you have no skills equipped, and sucky gear, right? When you're late game you have four skills equipped, and amazing gear, right? I think this is tied to how often the dimension shows up, AND I tested it out, and my experience confirms this. I'm not sure if it's the equipped gear, or the equipped skills, but I did try this out, and this is what I found.
My hour and a half of grinding was with four skills and good gear, and I didn't get even one dimension. I then unequipped all my skills (had zero skills equipped), and I played for a few minutes, and BAM I get a dimension run. One time, no joke, I got a dimension a minute and a half into my campaign. Obviously I didn't do well, because I had no skills and no gear. Then, I tried equipping a skill, and while it took a few minutes, I got another run. One time I had one skill equipped, and quite good gear (flamethrower :D), and then I got a dimension run. Because of all this, I personally believe that the frequency of dimension runs is directly tied to how many skills you have equipped. That being said, one time I had only one skill equipped, and I spent about 15 minutes without a dimension run, so it might literally be entirely random. However, I did find that my dimension runs went up drastically when I had less skills and less gear.

How do I last 20 waves?
Took me a good few serious tries. I have a few tips for you.
1. Your gear is a bit tricky for this achievement, for the reasons listed in how to get to the SynthWave world. What I did is I would start my mission with default gear, go to the shop and see if they have anything good (tomahawk is best, because it can pierce enemies and travels a good distance, which means easy explosions. Tomahawk is unlocked using RadRocks). If the shop doesn't have anything good, I restart. Once I get a good thing to purchase, I grind enough money to get it. Interestingly, I often would get to the SynthWave before getting enough money (for the reasons in the "how do I get there?" question above).
2. Spam melee. It blocks bullets, usually pushes enemies away from you, AND it slows the game down. You'll notice if you spam melee for a minute or so, and then stop, all of a sudden the game feels faster. That's because the melee actually slows things down a little. Just keep tapping the melee button, while holding down the shoot button. This way you'll swing, then shoot, then swing, then shoot, etc. This conserves your ammo, deflects bullets, and slows things down.
3. "360 Blowscope" saved me for this achievement. I would get surrounded by veggies, take a melee swing, and BOOM everything dies. Soooo helpful. If you manage to get a piercing ranged melee weapon, woah, easy explosions, and you'll have an easy time getting this achievement.
4. One of the devs said to me that if you fall off the edge of the level, that's gameover for your SynthWave run. This is why your hearts seem irrelevant at times. Just try not to fall off.

With all this in mind, you should be able to get to wave 20, and 100% the game.
Good luck fellow veggetable murderers!